// Numbas version: finer_feedback_settings {"name": "Luis's copy of Differentiation: Product rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"tags": ["algebraic manipulation", "calculus", "Calculus", "checked2015", "derivatives", "derivatives ", "differentiate a product", "differentiate polynomials", "differentiation", "elementary differentiation", "mas1601", "MAS1601", "polynomials", "product rule", "steps", "Steps"], "statement": "
Differentiate the following function $f(x)$ using the product rule.
", "name": "Luis's copy of Differentiation: Product rule", "metadata": {"notes": "\n \t\t31/07/2012:
\n \t\tAdded tags.
\n \t\tAdded description.
\n \t\tSteps problem to be addressed via an issue. Now resolved.
\n \t\tChecked calculation.OK.
\n \t\tImproved prompt display.
\n \t\tClicking on Show steps does not lose any marks.
\n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Differentiate $f(x) = x^m(a x+b)^n$.
"}, "ungrouped_variables": ["a", "s1", "b", "m", "n"], "parts": [{"prompt": "\n$\\displaystyle \\simplify[std]{f(x) = x ^ {m} * ({a} * x+{b})^{n}}$
\n$\\displaystyle \\frac{df}{dx}=\\;$[[0]]
\nClicking on Show steps gives you more information, you will not lose any marks by doing so.
\n ", "type": "gapfill", "marks": 0, "showCorrectAnswer": true, "gaps": [{"checkvariablenames": false, "showCorrectAnswer": true, "answer": "{m}x ^ {m-1} * ({a} * x+{b})^{n}+{n*a}x^{m} * ({a} * x+{b})^{n-1}", "vsetrange": [0, 1], "answersimplification": "std", "expectedvariablenames": [], "vsetrangepoints": 5, "type": "jme", "marks": 3, "checkingtype": "absdiff", "scripts": {}, "checkingaccuracy": 0.001, "showpreview": true}], "stepsPenalty": 0, "scripts": {}, "steps": [{"prompt": "The product rule says that if $u$ and $v$ are functions of $x$ then
\\[\\simplify[std]{Diff(u * v,x,1) = u * Diff(v,x,1) + v * Diff(u,x,1)}\\]
The product rule says that if $u$ and $v$ are functions of $x$ then
\\[\\simplify[std]{Diff(u * v,x,1) = u * Diff(v,x,1) + v * Diff(u,x,1)}\\]
For this example:
\n \n \n \n\\[\\simplify[std]{u = x ^ {m}}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {m}x ^ {m -1}}\\]
\n \n \n \n\\[\\simplify[std]{v = ({a} * x+{b})^{n}} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {n*a} * ({a} * x+{b})^{n-1}}\\]
\n \n \n \nHence on substituting into the product rule above we get:
\n \n \n \n\\[\\simplify[std]{Diff(f,x,1) = {m}x ^ {m-1} * ({a} * x+{b})^{n}+{n*a}x^{m} * ({a} * x+{b})^{n-1}}\\]
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